The Ising model with antiferromagnetic couplings on a family of generalized Sierpinski gaskets is studied using an exact technique developed recently by Stinchcombe [Phys. Rev. B 41 (1990) 2510]. A general expression of the zero-temperature entropy per spin, S(b)AFM (0), is given for general b, where b is a rescaling factor characterizing the fractals. Exact expressions for S(b)AFM (0) are derived for the cases b = 2, 3, 4, 5, 6. These expressions are evaluated numerically and results are compared to those obtained previously by numerical iterations of renormalization-group equations. The asymptotic behavior of S(b)AFM (0) in the limit of large b is discussed.